A-a- -B -b- -C -c- are the vertices of a triangle ABC and R -r- is any point in the plane of triangle ABC- then r-a-b- b-c- c-a- is always equal to Zero-a-b-c-a-b-c-None of these-

The correct option is B [a⃗ b⃗ c⃗] A vector perpendicular to the plane of A (a⃗), B (b⃗), C(c⃗) is (b⃗ a⃗) × (c⃗ a⃗) =a⃗×b⃗+b⃗×c⃗+c⃗×a⃗ Now for any poin ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard IX

Mathematics

Test for Coplanarity

A(a⃗), B (b⃗...

Question

$A (\to a), B (\to b), C (\to c)$ are the vertices of a triangle ABC and $R (\to r)$ is any point in the plane of triangle ABC, then $\to r . (\to a \times \to b + \to b \times \to c + \to c \times \to a)$ is always equal to

Zero

No worries! We‘ve got your back. Try BYJU‘S free classes today!

$[\to a \to b \to c]$

Right on! Give the BNAT exam to get a 100% scholarship for BYJUS courses

$- [\to a \to b \to c]$

No worries! We‘ve got your back. Try BYJU‘S free classes today!

None of these

No worries! We‘ve got your back. Try BYJU‘S free classes today!

Open in App

Solution

The correct option is B
$[\to a \to b \to c]$

A vector perpendicular to the plane of $A (\to a), B (\to b), C (\to c) i s (\to b - \to a) \times (\to c - \to a) = \to a \times \to b + \to b \times \to c + \to c \times \to a$
Now for any point $R (\to r)$ in the plane of A, B, C
$(\to r - \to a) . (\to a \times \to b + \to b \times \to c + \to c \times \to a) = 0 \Rightarrow \to r . (\to a \times \to b + \to b \times \to c + \to c \times \to a) - \to a . (\to a \times \to b + \to b \times \to c + \to c \times a) = 0 \Rightarrow \to r . (\to a \times \to b + \to b \times \to c + \to c \times \to a) = 0 + \to a . \to b \times \to c + 0 \Rightarrow \to r . (\to a \times \to b + \to b \times \to c + \to c \times \to a) = [\to a \to b \to c]$

Suggest Corrections

Similar questions

$A (\to a), B (\to b), C (\to c)$ are the vertices of a triangle ABC and $R (\to r)$ is any point in the plane of triangle ABC, then $\to r . (\to a \times \to b + \to b \times \to c + \to c \times \to a)$ is always equal to